Nominal model

In general, however, robust control system design uses an idealized, or nominal model of the plant Uncertainty in the nominal model is taken into account by... 

Nominal model-based observer can be extended to optimal LQG regulator... 

There are a number of general techniques suggested by the problem formulation. At the most detailed level of design, the design parameters need to be optimized in relation to performance criteria based on a nonlinear dynamic model. This points to a need for effective tools for dynamic optimization. At a more preliminary level in a hierarchy of techniques, it might be useful to evaluate steady-state performance or to carry out tests on achievable dynamic performance to eliminate infeasible options. Appropriate screening techniques are therefore needed. All these methods can use nominal models for initial analysis, but a full analysis should be based on design with uncertainty. 

We shall call this the nominal model . Comparing equation (24.3) with the standard simulation equation (2.22), it will be seen that the dependence of the vector function, f, on its constant parameters, a, has been made explicit in equation (24.3). 

Following the model-distortion idea described above, let us vary dynamically those parameters hitherto deemed constant. The values of the state variables will now be different from those of the nominal model, with the difference dependent on the extent of the variation in the parameters. Letting the new states be denoted by the n-dimensional vector, X, they will follow the equation... 

Inserting a parameter value of aj a p,j into the nominal model will produce the same output as applying a steady value Oj +< > to the distorted... 

Figure 24.2 Mean squared error versus parameter value for the nominal model and also for the distorted model when driven with a constant, non-optimal parameter value.

Using equation (24.25), the error associated with the ith output of the distorted model, which will be the same as the corresponding error for the nominal model, is... 

Using the procedure outlined above, we may generate a total of Ki > k inferred state histories , Z from the k original instrument outputs, z. We now sort the differential equations (DEs) defining the nominal model (equation (24.3)) into sets on the basis of the state variables they contain ... 

One of the primary motivations for feedback is to overcome model uncertainty. For the controller to work well in practice, it should be tuned to be robustly stabilizing. That is, the controller should not only stabilize the nominal model, but also all models within some uncertainty region that reflects how well the system has been identified. The areas of robust stability and robust performance are current topics of control research. To date, essentially none of this research has been applied to crystallization. 

This depends on the sensor configuration vector in A and the chosen parameters 0 > of the nominal model. The optimal sensor configuration given the nominal model is chosen as the one that minimizes this information entropy measure over all possible configurations. 

The optimal sensor configuration, obtained by minimizing Equation (3.70), depends on the designer s choice of the nominal model determined by the nominal parameter vector 0. One way to account for the uncertainty in the nominal model is to use a prescribed PDF p(0 C) for 0. In this case, the optimal sensor configuration becomes the one that minimizes the robust information entropy Eg/lHeiAlSf, 0, C)] which is a measure of the overall uncertainty in both 0 and 0 ... 

Consider next the incomplete mode shape measurements where only six sensors on the first, fourth, fifth, seventh, tenth and top floors are available. The results presented in Table 5.2 are based on five measured modes and show the initial values, final most probable values, standard deviations and COVs of the stiffness parameters, which are comparable to the COV of the modal data. Figure 5.1 shows the iterative history for the most probable values of the stiffness parameters, with convergence occurring in about 120 iterations. Again, the same set of nominal stiffness values is used so the nominal model overestimated the interstory stiffnesses by 100 to 200%. The parameters converge very quickly even for such an unsatisfactory set of initial values. The CPU time for 200 iterations is about 0.8 s with a conventional dual CPU 3.0 GHz personal computer under the MATLAB environment [171]. Figure 5.2 shows the comparison between the identified system mode shapes (solid lines) and the actual mode shapes (dashed lines) for the first five modes but the two sets of curves are on top of each other. Of course, it is no wonder that the mode shape components of the observed degrees of freedom are estimated better than the unobserved ones. 

To test alternative control strategies, Prett and Morari (1986) provide a linearized model, referred to as the Shell process, of a distillation tower to separate crude oil into fractions in a refinery. Part of the model describes the dynamics of the two top compositions as a function of the manipulated variables (the two top draw rates) and two key disturbances (the heat removal loads in pump-around streams used to remove heat and create intermediate reflux). For this example, it is sufficient to examine die matrices specific to the nominal model ... 

In the following, we will call nominal model the deterministic best-estimate model. This model is subject to uncertainties, the latter are supposed to be mainly of epistemic nature. We wiU consider here uncertainty due to imperfect knowledge of the mechanical properties of the turbogenerator and its interaction with the supporting structure in the machine hall. 

Let 0 denote the vector of component parameters i,..., 0, let y(f) be the vector of known measured outputs of the real system, (f) the vector of inputs into the system and into the nominal model of the system and wi (f) the vector of all switch states, i.e. m t) denotes the system mode. Assume that fault diagnosis has detected and isolated 1 as a single incipient fault starting at time instant to. Then... 

Alternatively, for off-line simulation, a behavioural model of areal system subject to faults and a reference model with nominal parameters can be coupled by residual sinks. Their outputs being ARRs residuals force the nominal model to adapt to the behaviour of the faulty system model. In this approach, the two coupled bond graphs are in integral causality. Advantage of this approach are that... 

A structural fault noted Fs corresponds to a new effort (or flow) source that causes a change in the structure of the model. Thus, the nominal model of the system is not conserved and its dynamic is altered by the presence of the fault. This difference between the system and the model generates an unbalance in the flow, mass and energy conservation laws, calculated from junctions 0 and 1 of the bond graph model. For example, a water leakage in the tank of Fig. 3.15b is a stfuctural fault. It can be modeled by a flow source Sf Yg. The model sfructure has changed from the bond graph model of the system without fault of Fig 3.15a. 

In a standard interconnection model (Fig. 7.22), the parameter uncertainties are separated from the nominal model and represented as feedback loops of internal variables ... 

The GARRs derived above can be separated into their nominal part and uncertain parts as and o 4), i.e., Rda = Rdsn as and Rd4 = Rd4 4. The threshold generator uses the contemporary inputs and the nominal model with uncertainties. 

Methodology. The methodology of the nonparametric probabilistic approach of uncertainties is as follows. (1) Developement of a finite element model of the designed elastoacoustic system. Such a model will be called the mean model (or the nominal model). (2) Construction of a reduced mean model from the mean model. (3) Construction of a stochastic reduced model from the reduced mean model using the nonparametric probabiUstic approach which allows the probability distribution of each random generalized matrix to be constructed. (4) Construction and convergence analysis of the stochastic solution. 

User defined This pattern is used in case the desired fault behavior can t be modeled using the fault library. The pattern allows then the custom modeling of this fault behavior in the same language of the implementation model. Accordingly, the fault is not injected later in the process in the nominal model but it... 

At runtime, the nominal model is injected with the faults. Additionally, an observer automaton for the analj d requirement is generated and injected in the model. The resulting overall model is finally translated to the VIS format and passed to the model checker. The analysis identifies all state sequences leading from the set of initial system states over the activation of faults to the observation of the violation of the functional requirement. These paths are the basis for computing the set of minimal cut-sets leading to the failure. 

A fundamental feature of the approach is model extension starting from a nominal model of the system, and a set of possible faults, the extension operator is able to generate a comprehensive description combining both the nominal and the faulty behaviours of the model. The SLIM language also allows for a comprehensive representation of partial observabihty, necessary to describe the actual sensing capabilities at the disposal of an on-hne monitoring system. 

The complementary sensitivity function for the nominal model and the PID controller is... 

Error Behavior. Error models are an extension to the specification of nominal models and are used to conduct safety, dependability and performability analyses. For modularity, they are defined separately from nominal specifications. Akin to nominal models, an error model is defined by its type and its associated implementations. 

An error model implementation (such as the one given in Listing 2) provides the structural details of the error model. It defines a (probabilistic) machine over the error states declared in the error model type. Transitions between states (fines 6-10) can be triggered by error events (fines 2-5), reset events, and error propagations. Error events are internal to the component they reflect changes of the error state caused by local faults and repair operations, and they can be annotated with occurrence distributions to express probabilistic error behavior. Moreover, reset events can be sent from the nominal model to the error model of the same component, trying to repair a fault that has occurred. Whether or... 

---